Proving gradient points in the direction of maximum increase

How do we prove that the gradient points in the direction of the maximum increase? Would it be enough to simply state that the gradient is just the derivates of a function w.r.t all the variables a function depends upon. Since the derivative of a term w.r.t a certain variable gives the maximum increase of that term and since in gradient not only do we account for all possible terms and all the variables upon which a term depends upon but also includes the direction, so it is logical to conclude that the gradient points in the direction of the maximum increase.

Note: I do hope that the above proof is adequate but somehow feel that the proof has to be mathematical. Could someone tell me whether or not my intuition is correct?

the answer depends on your definition of the gradient. if you think the gradient is the vector that dots with a direction vector to give the directional derivative, then the previous explanation is all you need.

I you think the gradient is a vector whose entries are the partial derivatives, then you need to make sense out of that a bit more. i.e. you need the chain rule to get the previous statement.

the answer depends on your definition of the gradient. if you think the gradient is the vector that dots with a direction vector to give the directional derivative, then the previous explanation is all you need.

I you think the gradient is a vector whose entries are the partial derivatives, then you need to make sense out of that a bit more. i.e. you need the chain rule to get the previous statement.

That is why I asked OP what, if anything, he found lacking.

However, since he had already posted other threads in which he showed familiarity with the concept of the directional derivative, I assumed he was also familiar with how that is derived.
Therefore, I did not address that point explicitly.